2 Answers
2

It depends what you’re trying to say. If you want to refer to a public project that’s well known, you need ‘the well-known public project’. If, on the other hand, the project in question is not necessarily a public one, but one which is nevertheless well known to the public, then it would be grammatical to say ‘the publicly well-known project’. Grammatical, but not very likely. What you’d be more likely to hear would be ‘the project well known to the public’.